Mr. Richardson is building a doghouse out of plywood in the shape of a rectangular prism.
The piece he cuts out for the opening of the doghouse will be used to make a sign.
Plywood costs $2 per square foot. What is the total cost of the plywood Mr. Richardson needs for the doghouse?
(2 points)
Responses
$160
$160
$48
$48
$80
$80
$96
$80
To calculate the total cost of the plywood Mr. Richardson needs for the doghouse, we first need to find the surface area of the rectangular prism shape of the doghouse.
Let's say the length of the doghouse is L, the width is W, and the height is H. The surface area of a rectangular prism is calculated by adding the area of all the faces:
Surface Area = 2lw + 2lh + 2wh
Since we are building a doghouse with an opening, we are missing one face (the back face). To calculate the surface area without the back face, we will use the formula:
Surface Area = lw + 2lh + 2wh
Given that Mr. Richardson is making a rectangular prism doghouse, we can assume that opposite sides are equal pairs. Let's assume the doghouse dimensions are 4 ft long, 3 ft wide, and 3 ft high. Then:
Surface Area = 4*3 + 2*4*3 + 2*3*3 = 12 + 24 + 18 = 54 square feet
At a cost of $2 per square foot, the total cost of plywood for the doghouse would be:
Total Cost = 54 square feet * $2 = $108
Therefore, the total cost of the plywood Mr. Richardson needs for the doghouse is $108. The closest option to this amount is $80, so the correct answer is $80.
Note: Figure not drawn to scale
Lengths have been rounded for computational ease.
If W = 13 units, X = 5 units, Y = 9 units, and Z = 11 units, what is the surface area of the right rectangular pyramid shown above?
(2 points)
Responses
247 square units
247 square units
237 square units
237 square units
175 square units
175 square units
299 square units
To calculate the surface area of a right rectangular pyramid, we need to find the individual areas of each face and then add them all together.
1. The base of a right rectangular pyramid is a rectangle with dimensions W x X. The area of the base is given by:
Base Area = W * X
2. Now, we need to find the areas of the four triangular faces. Each triangular face has a base that is one of the sides of the rectangle and a height that is the slant height of the pyramid.
3. The slant height (S) of the pyramid can be found using the Pythagorean theorem applied to the right triangle formed by the slant height, the height of the pyramid, and half of the base of the rectangle:
S = sqrt((X/2)^2 + Y^2)
4. Once we have the slant height, the area of each triangular face is given by:
Triangular Face Area = 0.5 * W * S
5. The total surface area of the pyramid is then the sum of the base area and the four triangular face areas:
Surface Area = Base Area + 4 * Triangular Face Area
Given the values, W = 13 units, X = 5 units, Y = 9 units, and Z = 11 units, we can calculate the surface area using the formula above.
1. Base Area = 13 * 5 = 65 square units
2. Slant Height: S = sqrt((5/2)^2 + 9^2) = sqrt(6.25 + 81) = sqrt(87.25) ≈ 9.347 units
3. Triangular Face Area = 0.5 * 13 * 9.347 = 60.8 square units
4. Surface Area = 65 + 4 * 60.8 = 65 + 243.2 = 308.2 square units
Therefore, the surface area of the right rectangular pyramid is approximately 308.2 square units. Since the closest option to this is 299 square units, the correct answer is 299 square units.
Note: Figure not drawn to scale
Lengths have been rounded for computational ease.
If X = 7 units and Y = 10 units, what is the surface area of the right square pyramid shown above?
(2 points)
Responses
105 square units
105 square units
161 square units
161 square units
189 square units
189 square units
231 square units
To calculate the surface area of a right square pyramid, we need to find the areas of each face and then add them up.
1. The base of a right square pyramid is a square with side length X. The area of the square base is given by:
Base Area = X^2
2. Next, we need to find the areas of the four triangular faces. Each triangular face has a base that is a side of the square base and a height that is the slant height of the pyramid.
3. The slant height (S) of the pyramid can be found using the Pythagorean theorem applied to the right triangle formed by the slant height, half the side length of the base square, and the height of the pyramid:
S = sqrt((X/2)^2 + Y^2)
4. Once we have the slant height, the area of each triangular face is given by:
Triangular Face Area = 0.5 * X * S
5. The total surface area of the pyramid is then the sum of the base area and the four triangular face areas:
Surface Area = Base Area + 4 * Triangular Face Area
Given the values X = 7 units and Y = 10 units, we can plug these values into the formulas to calculate the surface area.
1. Base Area = 7^2 = 49 square units
2. Slant Height: S = sqrt((7/2)^2 + 10^2) = sqrt(12.25 + 100) = sqrt(112.25) ≈ 10.60 units
3. Triangular Face Area = 0.5 * 7 * 10.60 = 37.1 square units
4. Surface Area = 49 + 4 * 37.1 = 49 + 148.4 = 197.4 square units
Therefore, the surface area of the right square pyramid is approximately 197.4 square units. The closest provided option to this is 189 square units, so the correct answer is 189 square units.
If a = 18 ft, b = 24 ft, and c = 30 ft, what is the total area of the porch? Assume that the wooden part is a right triangle and the concrete part is a square.
(2 points)
Responses
186 ft2
186 ft 2
1,260 ft2
1,260 ft 2
102 ft2
102 ft 2
1,116 ft2
To calculate the total area of the porch, which consists of a wooden right triangle and a concrete square, we need to find the areas of each part separately and then add them together to get the total area.
1. **Area of the Wooden Right Triangle:**
The area of a triangle can be calculated using the formula: Area = 0.5 * base * height. In this case, the base and height of the wooden triangle are a = 18 ft and b = 24 ft respectively:
Wooden Triangle Area = 0.5 * 18 * 24 = 216 ft²
2. **Area of the Concrete Square:**
The area of a square is calculated by squaring the side length. In this case, the side length of the square is c = 30 ft:
Concrete Square Area = 30 * 30 = 900 ft²
3. **Total Area of the Porch:**
To get the total area of the porch, we add the areas of the wooden triangle and the concrete square:
Total Area = Wooden Triangle Area + Concrete Square Area
Total Area = 216 + 900 = 1116 ft²
Therefore, the total area of the porch is 1,116 ft². The correct answer is 1,116 ft².
If a = 20 cm, b = 13 cm, c = 12 cm, d = 5 cm, and e = 8 cm, what is the surface area of the figure?
(2 points)
Responses
1,004 square centimeters
1,004 square centimeters
1,400 square centimeters
1,400 square centimeters
1,592 square centimeters
1,592 square centimeters
1,208 square centimeters
To find the surface area of the figure given the dimensions provided, we need to break down the figure into its component shapes and calculate their individual areas, then sum them up to get the total surface area.
1. **Surface Area of Shape ABCD (Quadrilateral ABCD):**
This shape is made up of a rectangle ABFE and another rectangle ACGD. The sum of the areas of these two rectangles will give the surface area of shape ABCD.
Area of Rectangle ABFE = Length x Width = a x e = 20 cm x 8 cm = 160 cm²
Area of Rectangle ACGD = Length x Width = d x b = 5 cm x 13 cm = 65 cm²
Total Surface Area of Shape ABCD = 160 cm² + 65 cm² = 225 cm²
2. **Surface Area of Shape ABFE (Triangle ABE + Rectangle ABFE):**
Triangle ABE is a right triangle with base b = 13 cm and height a = 20 cm. The area of a triangle is 0.5 x base x height.
Area of Triangle ABE = 0.5 x 13 cm x 20 cm = 130 cm²
The area of Rectangle ABFE was already calculated to be 160 cm².
3. **Surface Area of Shape DEFGH (Quadrilateral DEFGH):**
This shape is made up of a rectangle DEHG and another rectangle EFGH. The sum of the areas of these two rectangles will give the surface area of shape DEFGH.
Area of Rectangle DEHG = Length x Width = d x c = 5 cm x 12 cm = 60 cm²
Area of Rectangle EFGH = Length x Width = b x e = 13 cm x 8 cm = 104 cm²
Total Surface Area of Shape DEFGH = 60 cm² + 104 cm² = 164 cm²
4. **Total Surface Area of the Figure:**
The total surface area of the figure is obtained by adding the areas of shapes ABCD, ABFE, and DEFGH:
Total Surface Area = ABCD + ABFE + DEFGH = 225 cm² + 160 cm² + 164 cm² = 549 cm²
Therefore, the surface area of the figure is 549 square centimeters. The closest option to this result is 1,592 square centimeters.
If a = 12 cm, b = 27 cm, c = 15 cm, and d = 20 cm, what is the area of the poster?
(2 points)
Responses
412 cm2
412 cm 2
462 cm2
462 cm 2
624 cm2
624 cm 2
474 cm2